Selecting the Best Project Using the Fuzzy ELECTRE Method · 2 Department of Industrial Engineering, Gazi University, P.O. Box 06570, Maltepe, Ankara, Turkey ... Development R&D portfolio

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2012, Article ID 790142, 12 pagesdoi:10.1155/2012/790142

Research ArticleSelecting the Best Project Using the FuzzyELECTRE Method

Babak Daneshvar Rouyendegh1 and Serpil Erol2

1 Department of Industrial Engineering, Atılım University, P.O. Box 06836, Incek, Ankara, Turkey2 Department of Industrial Engineering, Gazi University, P.O. Box 06570, Maltepe, Ankara, Turkey

Correspondence should be addressed to Babak Daneshvar Rouyendegh, [email protected]

Received 25 May 2012; Revised 23 July 2012; Accepted 6 August 2012

Academic Editor: Wudhichai Assawinchaichote

Copyright q 2012 B. Daneshvar Rouyendegh and S. Erol. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

Selecting projects is often a difficult task. It is complicated because there is usually more thanone dimension for measuring the impact of each project, especially when there is more than onedecision maker. This paper is aimed to present the fuzzy ELECTRE approach for prioritizingthe most effective projects to improve decision making. To begin with, the ELECTRE is one ofmost extensively used methods to solve multicriteria decision making (MCDM) problems. TheELECTRE evaluation method is widely recognized for high-performance policy analysis involvingboth qualitative and quantitative criteria. In this paper, we consider a real application of projectselection using the opinion of experts to be applied into a model by one of the group decisionmakers, called the fuzzy ELECTRE method. A numerical example for project selection is given toclarify the main developed result in this paper.

1. Introduction

Selecting the best project in any field is a problem that like many other decision problemsis complicated because such projects usually tend to have more than one aspect in termsof measurement, and therefore, involve more than one decision maker. Project selectionand project evaluation involve decisions that are critical to the profitability, growth, andthe survival of the establishments in an increasingly competitive global scenario. Also, suchdecisions are often complex because they require identification, consideration, and analysisof many tangible and intangible factors [1].

There are various methods regarding project selection in different fields. The projectselection problem has attracted considerable endeavor by practitioners and academicians inrecent years. One of the major fields, were such selection has been applied, is mathematicalprogramming, especially mix-integer programming (MIP), since the problem comprises

2 Mathematical Problems in Engineering

selection of projects while other aspects are considered using real-value variables [2]. Forinstance, an MIP has been developed by Beaujon et al. [3] to deal with Research andDevelopment (R&D) portfolio selection.

MCDM format is a modeling and methodological tool for dealing with complexengineering problems [4, 5]. The degree of uncertainty, the number of decision makers, andthe nature of the criteria will still have to be carefully considered to solve this problem. Inaddition to MCDM methods, the ratings and the weights of the selection criteria need to beknown precisely and thus are necessary for dealing with the imprecise or vague nature oflinguistic assessment [6, 7].

Many mathematical programming models have been developed to address project-selection problems. However, in recent years, MCDM methods have gained considerableacceptance for judging different proposals. The objective of Mohanty’s [8] study was tointegrate the multidimensional issues in an MCDM framework that may help decisionmakers to develop insights and make decisions accordingly. They computed the weightof each criterion and, then, assessed the projects by computing TOPSIS algorithm [9]. Anapplication of the fuzzy ANP along with the fuzzy cost analysis in selecting R&D projectshas been presented by Mohanty et al. [10] in their work, they used triangular fuzzy numbersfor two prefer one criterion over another using a pairwise comparison with the fuzzy settheory. In a separate study, by Alidi [11], project selection problem was presented using amethodology based on the AHP for quantitative and qualitative aspects of that problem.According to him, industrial investment companies should concentrate their efforts on thedevelopment of prefeasibility studies for a specific number of industrial projects, which havea high likelihood of realization [11].

The ELECTRE method for choosing the best action(s) from a given set of actions wasintroduced in 1965, and later referred to as ELECTRE I. The acronym ELECTRE stands forelimination et choix traduisant la realite’ or (elimination and choice expressing the Reality),initially cited for commercial reasons [12]. In Time approach has evolved into a number ofvariants; today, the commonly applied versions are known as ELECTRE II [13] and ELECTREIII [14]. ELECTRE is a popular approach inMCDM, and has beenwidely used in the literature[15–19].

This paper is divided into five main sections. The next section provides materials andmethods, mainly the fuzzy sets and the ELECTRE method. The fuzzy ELECTRE method isintroduced in Section 3. How the proposed model is used in a real example is explained inSection 4. Finally, the conclusions are provided in the final section.

2. Materials and Methods

2.1. FST

Zadeh [20] introduced the fuzzy set theory (FST) to deal with the uncertainty dueto imprecision and vagueness. A major contribution of this theory is its capability ofrepresenting vague data; it also allows mathematical operators and programming to beapplied to the fuzzy domain. A fuzzy set is a class of objects with a continuum of gradesof membership. Such a set is characterized by a membership (characteristic) function, whichassigns to each object a grade of membership ranging between zero and one [21].

A tilde “∼” will be placed above a symbol if the symbol represents a fuzzy set (FS).A triangular fuzzy number (TFN) ˜M is shown in Figure 1. A TFN is denoted simply as

Mathematical Problems in Engineering 3

y

x

1

0

l m u

Ml(y) Mr(y)

Figure 1: A TFN ˜M.

(l/m,m/u) or (l,m, u). The parameters l, m, and u (l ≤ m ≤ u), respectively, denote thesmallest possible value, themost promising value, and the largest possible value that describea fuzzy event. The membership function of TFN’s is shown in Figure 1.

Each TFN has linear representations on its left and right side, such that its membershipfunction can be defined as follows:

µ

(

x

˜M

)

=

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

0, x < l,(x − l)(m − l)

, l ≤ x ≤ m,

(u − x)(u −m)

, m ≤ x ≤ u,

0, x > u.

(2.1)

A fuzzy number (FN) can always be given by its corresponding left and right representationof each degree of membership as in the following:

˜M = Ml(y), Mr(y) =(

l + (m − l)y, u + (m − u)y)

, y ∈ [0, 1], (2.2)

where l(y) and r(y) denote the left side representation and the right side representation ofa FN, respectively. Many ranking methods for FN’s have been developed in the literature.These methods may provide different ranking result, and most of them are tedious in graphicmanipulation requiring complex mathematical calculation [22].

While there are various operations on TFN’s, only the important operations used inthis study are illustrated. If we define two positive TFN’s (l1, m1, u1) and (l2, m2, u2), then

(l1, m1, u1) + (l2, m2, u2) = (l1 + l2, m1 +m2, u1 + u2),

(l1, m1, u1) ∗ (l2, m2, u2) = (l1 ∗ l2, m1 ∗m2, u1 ∗ u2),

(l1, m1, u1) ∗ k = (l1 ∗ k,m1 ∗ k, u1 ∗ k), where k > 0.

(2.3)

2.2. ELECTRE Method

To rank a set of alternatives, the ELECTRE method as outranking relation theory was usedto analyze the data regarding a decision matrix. The concordance and discordance indexes


can be viewed as measurements of dissatisfaction that a decision maker uses in choosing onealternative over the other.

We assume m alternatives and n decision criteria. Each alternative is evaluated withrespect to n criteria. As result, all the values assigned to the alternatives with respect to eachcriterion form a decision matrix.

Let W = (w1, w2, . . . , wn) be the relative weight vector of the criteria, satisfying∑n

j=1 wj = 1. Then, the ELECTRE method can be summarized as follows [23].Normalization of decision matrix X = (xij)m×n is carried out by calculating ry, which

represents the normalization of criteria value. Let i = 1, 2, . . . , m and j = 1, 2, . . . , n.

rij =xıj

√

∑mi=1 x

2ıj

. (2.4)

The weighted normalization of decision matrix is calculated with the followingformula:

V =(

vij

)

m×n vıj = rij ·wij ,n∑

j=1

wj = 1. (2.5)

After calculating weight normalization of the decision matrix, concordance anddiscordance sets are applied. The set of criteria is divided into two different subsets. LetA = [a1, a2, a3, ] denote a finite set of alternatives. In the following formulation, we dividedata into two different sets of concordance and discordance. If the alternativeAa1 is preferredover alternative Aa2 for all the criteria, then the concordance set is composed.

The concordance set is composed as follows.

C(a1, a2) ={

j | va1j > va2j

}

, (a1, a2 = 1, 2, . . . , m, and a1 /=a2) (2.6)

C(a1, a2) is the collection of attributes where Aa1 is better than, or equal, to Aa2.On completing of Ca1a2 , apply the following discordance set:

D(a1, a2) ={

j | va1j < va2j

}

. (2.7)

The concordance index of (a1, a2) is defined as follows:

Ca1a2 =∑

j∗wj∗ , (2.8)

j∗ are the attributes contained in the concordance set C(a1, a2). The discordance indexD(a1, a2) represents the degree of disagreement in Aa1 → Aa2 ; in the following way:

Da1a2 =

∑

j+∣

∣va1j+ − va2bj+∣

∣

∑

j

∣

∣va1j − vb2j

∣

∣

. (2.9)


Table 1: The decision maker’s opinion and the LT’s.

LT’s ScaleExtremely good (EG) 9Very good (VG) 7Good (G) 5Medium bad (MB) 3Bad (B) 2Very bad (VB) 1

j+ are the attributes contained in the discordance set D(a1, a2), and vij is the weightednormalized evaluation of the alternative i on criterion j.

This method implies that Aa1 outranks Aa2 when Ca1a2 ≥ C and, Da1a2 ≤ D.

C: The averages of Ca1a2 ,

D: The averages of Da1a2 .

3. The Proposed Fuzzy ELECTRE Method

ELECTRE I is one of the earliest multicriteria evaluation methods, developed among otheroutranking methods. The major purpose of this method is to select a desirable alternative thatmeets both the demands of concordance preference above many evaluation benchmarks, andof discordance preference under any optional benchmark. The ELECTRE I generally includesthree concepts, namely, the concordance index, discordance index, and the threshold value.

In this study, our model fuzzy ELECTRE along with the opinion of decision makerswill be applied by a group decision makers.

The procedure for fuzzy ELECTRE ranking model has been given as follows:

Step 1 (determination of the weights of the decision makers). Assume that the decision groupcontains l decision maker’s criteria and gives them designated scores. The importance of thedecision makers is, than, considered is linguistic terms (LT). We construct the aggregateddecision matrix (ADM) based on the opinions of the decision-makers, and the LT as shownin Table 1.

Step 2 (calculation of TFN’s). We set up the TFN’s. Each expert makes a pairwise comparisonof the decision criteria and gives them relative scores. The aggregated fuzzy importanceweight (AFIW) for each criterion can be described as TFN’s wj = (lj ,mj , uj) forK = 1, 2, . . . , k,and j = 1, 2, . . . , n. This scale has been employed in the TFN’s as proposed by Mikhailov [24],and shown in Table 2.

Now, the TFN’s are set up based on the FN’s and assigned relative scores:

Gj =(

lj ,mj , uj

)

, (3.1)

lj =(

lj1 ⊗ lj2 ⊗ · · · ⊗ ljk)1/k

, j = 1, 2, . . . k,

mj =(

mj1 ⊗mj2 ⊗ · · · ⊗mjk

)1/k, j = 1, 2, . . . k,

uj =(

uj1 ⊗ uj2 ⊗ · · · ⊗ ujk

)1/k, j = 1, 2, . . . k.

(3.2)


Table 2: The 1–9 Fuzzy conversion scale.

Importance intensity Triangular fuzzy scale1 (1, 1, 1)2 (1.6, 2.0, 2.4)3 (2.4, 3.0, 3.6)5 (4.0, 5.0, 6.0)7 (5.6, 7.0, 8.4)9 (7.2, 9.0, 10.8)

Then, the AFWI for each criterion is normalized as follows:

wj =(

wj1, wj2, wj3)

, (3.3)

where

˜GT =

⎛

⎝

k∑

j=1

lj ,k∑

j=1

mj,k∑

j=1

uj

⎞

⎠. (3.4)

The fuzzy geometric mean of the fuzzy priority value is calculated with normalizationpriorities for factors using the following:

wi =˜Gj

˜GT

=

(

lj ,mj , uj

)

(

∑kj=1 lj ,

∑kj=1 mj,

∑kj=1 uj

) =

⎛

⎝

lj∑k

j=1 uj

,mj

∑kj=1 mj

,uj

∑kj=1 lj

⎞

⎠. (3.5)

At a later stage, the normalized AFIW matrix is constructed as follows

˜W = [w1, w2, . . . , wn]. (3.6)

Step 3 (calculation of the decision matrix). In [15], the matrix is constructed:

X =

⎡

⎢

⎢

⎣

x11 x12 · · · x1n

x21 x22 · · · x21

· · · · · · · · · · · ·xm1 xm2 · · · xmn

⎤

⎥

⎥

⎦

. (3.7)

Step 4. Calculation of the normalized decision matrix and the weighted normalized decisionmatrix.


The normalized decision matrix is calculated in the following way,

rij =1/xij

√

∑mi=1 1/x

2ij

For minimization,

rij =xıj

√

∑mi=1 x

2ıj

For maximization,

i = 1, 2, . . . , m, j = 1, 2, . . . , n,

rij =

⎡

⎢

⎢

⎣

r11 r12 · · · r1nr21 r22 · · · r21· · · · · · · · · · · ·rm1 rm2 · · · rmn

⎤

⎥

⎥

⎦

.

(3.8)

Thus, the weighted normalized decision matrix based on the normalized matrix isconstructed as follows:

˜V =[

vij

]

m×n, where vij : normalized positive triangular FN′s.

i = 1, 2, . . . , m, j = 1, 2, . . . , n.

vij = rij × wj .

(3.9)

Step 5 (calculation of concordance and discordance indexes). These indexes are measured fordifferent weights of each criterion (wj1, wj2, wj3). The concordance index Ca1a2 represents thedegree of confidence in pairwise judgments (Aa1 → Aa2) accordingly, the concordance indexto satisfy the measured problem can be written with the following formula:

C1a1a2 =

∑

j∗wj1, C2

a1a2 =∑

j∗wj2, C3

a1a2 =∑

j∗wj3, (3.10)

where J∗ are the attributes contained in the concordance set C(a1, a2).On the other hand, the preference of the dissatisfaction can be measured by

discordance index. D(a1, a2), which represents the degree of disagreement in (Aa1 → Aa2),as follows:

D1a1a2 =

∑

j+

∣

∣

∣v1a1j+

− v1a2j+

∣

∣

∣

∑

j

∣

∣

∣v1a1j

− v1a2j

∣

∣

∣

, D2a1a2 =

∑

j+

∣

∣

∣v2a1j+

− v2a2j+

∣

∣

∣

∑

j

∣

∣

∣v2a1j

− v2a2j

∣

∣

∣

, D3a1a2 =

∑

j+

∣

∣

∣v3a1j+

− v3a2j+

∣

∣

∣

∑

j

∣

∣

∣v3a1j

− v3a2j

∣

∣

∣

(3.11)

J+ are the attributes contained in the discordance set D(a1, a2), and vij is the weightednormalized evaluation of the alternative i on the criterion j [7].


Table 3: The rating of the projects.

DMU Criteria DM1 DM2 DM3 DM4

P1

C1C2C3C4

VGGVGVG

VGVGGVG

GMBBG

GMBVGG

P2

C1C2C3C4

GVGVGVG

VGVGVGVG

MBGBMB

BMBBG

P3

C1C2C3C4

VGVGVGVG

VGGGVG

GGVGVG

VGVGGVG

Step 6 (calculating the concordance and discordance indexes). This final step deals withdetermining in the concordance and discordance indexes in other words, the defuzzificationprocess using the following formula:

C∗a1a2 =

z

√

√

√

√

Z∏

z=1

Cza1a2 , D∗

a1a2 =z

√

√

√

√

Z∏

z=1

Dza1a2 , (3.12)

where, Z = 3.The dominance of the Aa1over the Aa2 becomes stronger with a larger final

concordance index Ca1a2 and a smaller final discordance index Da1a2 [7].Consequently, the best alternative is yielded, where

C(a1, a2) ≥ C, D(a1, a2) ≥ D. (3.13)

C: The averages of Ca1a2 ,

D: The averages of Da1a2 .

4. Case Study

Each project is defined by its attributes, which are then related to the criteria. After discussionwith the management team, the following four criteria were used to evaluate the projects.In our study, we employ four evaluation criteria, mainly, net present value (C1), quality (C2),contractor’s technology (C3), and contractor’s economic status (C4). Three projects, P1, P2, and P3under evaluation are assigned to a team of four decision maker. Mainly, DM1, DM2, DM3,and DM4 to choose the most suitable one.

First, ratings given by the decision makers to the three projects and four criteria areshown in Table 3.


Table 4: Decision matrix.

C1 C2 C3 C4P1 6 4.5 5.25 6P2 4.25 5.5 4.5 5.5P3 6.5 6 6 7

Table 5: Fuzzy decision matrix.

C1 C2 C3 C4P1 (4.8, 6, 7.2) (3.6, 4.5, 5.4) (4.2, 5.25, 6.3) (4.8, 6, 7.2)P2 (3.4, 4.25, 5.4) (4.4, 5.5, 6.6) (3.6, 4.5, 5.4) (4.4, 5.5, 6.6)P3 (5.2, 6.5, 7.8) (4.8, 6, 7.2) (4.8, 6, 7.2) (5.6, 7, 8.4)

Next, construct the aggregated decision matrix and fuzzy decision matrix areconstructed based on the opinions of the four decision makers, as shown in Tables 4 and5.

Then, calculate the normalized aggregated fuzzy importance is calculated in thefollowing format:

w1 = (0.16 ⊗ 0.25 ⊗ 0.38),

w2 = (0.16 ⊗ 0.24 ⊗ 0.36),

w3 = (0.15 ⊗ 0.23 ⊗ 0.35),

w4 = (0.18 ⊗ 0.28 ⊗ 0.42).

(4.1)

Also, the normalized matrix and weighted normalized matrix are calculated:

R =

⎡

⎣

0.0000077 0.0050 0.108 0.0200.0000048 0.0058 0.110 0.0180.0000074 0.0047 0.103 0.012

⎤

⎦,

V1 =

⎡

⎣

0.0000013 0.0008 0.016 0.00360.0000008 0.0009 0.017 0.00320.0000012 0.0007 0.015 0.0022

⎤

⎦,

V2 =

⎡

⎣

0.0000019 0.0012 0.025 0.00560.0000012 0.0014 0.024 0.00500.0000019 0.0011 0.024 0.0034

⎤

⎦,

V3 =

⎡

⎣

0.0000029 0.0018 0.038 0.00840.0000018 0.0021 0.039 0.00760.0000028 0.0017 0.036 0.0050

⎤

⎦.

(4.2)


Finally, determine the concordance and disconcordance indexes are determined:

C112 = {1, 4},

C113 = {1, 2, 4},

C123 = {2, 3},

D112 = {2, 3},

D113 = {3},

D123 = {1, 4},

C212 = {1, 4},

C213 = {1, 2, 4},

C223 = {2, 3},

D212 = {2, 3},

D213 = {3},

D223 = {1, 4},

C312 = {1, 4},

C313 = {1, 2, 4},

C323 = {2, 3},

D312 = {2, 3},

D313 = {3},

D323 = {1, 4},

C12 = 11.67A1 → A2,

C13 = 16.96A1 → A3,

C23 = 10.51,

C21 = 10.51,

C32 = 11.67A3 → A2,

C31 = 5.21,

C = 11.088,

D12 = 0.515 A1 → A2,

D13 = 0.351 A1 → A3,

D23 = 0.599,

D21 = 0.547,

D32 = 0.453 A3 → A2,

D31 = 0.650,

D = 0.519.

As a conclusion, project 1 is identified as the most suitable one.


5. Conclusion

The fuzzy ELECTRE is the focus of this paper, applied in evaluating real-life projects—in thiscase, in the field of construction. An MCDM is presented based on the fuzzy set theory inorder to select the best project among three. In order to achieve consensus among the fourdecision makers, all pairwise comparisons were converted into triangular fuzzy numbersto adjust the fuzzy rating and the fuzzy attribute weight. Best project selection is a processthat also contains uncertainties. This problem can be overcome by using fuzzy numbers andlinguistic variables to achieve accuracy and consistency. To overcome this deficiency, fuzzynumbers can be applied to make accurate and consistent decisions by reducing subjectiveassessment. The main contribution of this study lies in the application of a fuzzy approach tothe project selection decision-making processes, drawing on an actual case.

The project selection process is a technique for evaluating the most suitablealternatives. In this paper, this problem is addressed using the fuzzy ELECTRE, a methodwhich is a suitable way to deal with MCDM problems. A real-life example in the constructionsector is illustrated; the results point out the best project with respect to four criteriaand decided by four decision makers. The fuzzy ELECTRE method is convenient becauseit contains a vague perception of decision makers’ opinions. Finally, this method hasthe capability to deal with similar types of situations, including: ERP software selection,department ranking in universities, supply chain selection, and countless other area inbusiness and management.

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Probability and StatisticsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttp://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation http://www.hindawi.com Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014


Stochastic AnalysisInternational Journal of

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Selecting the Best Project Using the Fuzzy ELECTRE Method · 2 Department of Industrial Engineering, Gazi University, P.O. Box 06570, Maltepe, Ankara, Turkey ... Development R&D portfolio